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CharlesIn Shirley Jackson's 1948 short story, "Charles," the main character Laurie is proud of the fictional Charles's behavior. Laurie invents the character of Charles on his first day of kindergarten....

The Magician's NephewThe first connection that comes to mind between vanity and being a magician is power. Someone who believes they are better than everyone else is also likely to believe they should be able to have...

Lord of the FliesThroughout the majority of the novel, Ralph is a proponent for civility and views the savages with contempt. After the boys agree to hunt the beast, Ralph joins the hunters as they explore the...

RikkiTikkiTaviI'm not sure if the question is asking for a verbal threat that Rikkitikki makes to Nag and Nagaina, or if the question is asking about how Rikkitikki is a threat in general. In general,...

HistoryGood question. There are two very different answers to this question. The first is that the deep structure of myths is built into the way we tell stories. Some scholars have argued it is built into...

HistoryImperialism refers to when a country controls another country politically and economically, sometimes against its will. The United States became imperialistic in the late 1800s. We manufactured a...

HistoryOn his second voyage to the New World, having already visited Hispaniola and the south of Cuba on his first, Columbus sought to continue his exploration in the Caribbean. He brought with him a...

All Summer in a DayI would say that the resolution of the story is that there is no resolution! This might seem to not make sense, but the story ends on a kind of cliffhanger: Margot is locked in a closet, and misses...

The Most Dangerous GameGeneral Zaroff is Connell's antagonist in "The Most Dangerous Game." As villains normally do, Zaroff has a different perspective towards life and society than the rest of the world does. For...

SeedfolksPaul Fleishman's novel Seedfolks is a series of character vignettes centered around a young girl named Kim who plants seeds in a vacant lot in the neighborhood. This begins a community garden in...

The Outcasts of Poker FlatThe obvious answer to this question would be Mr. Oakhurst. He's the one who leads the group out of Poker Flat and into the mountains, hoping to speed them along before the weather hits. He's...

MathTo solve, let's apply the least common multiple (LCM) of a set of numbers. Pizza is served every sixth day. To get the days in which it will be served, take the multiples of 6. The multiples of 6...

The Hitchhiking GameFirst, it's important to determine the character of the young man before we begin our letter. Identifying the young man's temperament and personalitytype will help us to stay as true to character...

The Catcher in the RyeIt's interesting to view Jane's checker playing as a metaphor for her approach to life. Kings in the back row might stand for selfdiscipline, reserve, or a hidden ally. One thing that Holden...

MathGiven the equations y=0, x=6 and `y=\frac{x}{\sqrt{x+3}}` We have to find the area bounded by the three equations. The graph is shown below: The yellow region is the bounded area. Point (6,2) is...

MathGiven to solve , `int 4/(csc(theta)cot(theta)) d theta` just for the convenience let` x= theta` so,` int 4/(csc(theta)cot(theta)) d theta` =`int 4/(csc(x)cot(x)) d x` =`int...

Math`int (sin sqrt theta)/sqrt theta d theta` To solve, apply usubstitution method. `u=sqrt theta` `u= theta ^(1/2)` `du = 1/2 theta^(1/2) d theta` `du = 1/(2theta^(1/2))d theta` `du =1/(2 sqrt...

MathGiven to solve , `int cos(theta) / (1+cos(theta)) d theta` just for easy solving let `x=theta ` so the equation is given as `int cos(x) / (1+cos(x)) d x ` (1) let `u= tan(x/2)` ,=> then...

Math`int ( sin (theta))/(32cos(theta)) d theta` To solve, apply usubstitution method. `u = 32cos (theta)` `du = 2*(sin (theta)) d theta` `du = 2sin (theta) d theta` `1/2du= sin (theta) d theta`...

RikkiTikkiTaviAt first Nag introduces himself to RikkiTikki in a way designed to strike fear into the mongoose's heart and to make Rikki feel like a young thing. But Rikki's mongoose nature won't allow him to...

MathBasis (n=1) We will use integration by parts `int u dv=uvint v du` `int_0^infty xe^x dx=[u=x,dv=e^x dx],[du=dx,v=e^x]=` `xe^x_0^infty+int_0^infty e^x dx=(xe^xe^x)_0^infty=` `lim_(x...

Math`int_0^infty sin(x/2)dx=` Substitute `u=x/2` `=>` `du=dx/2` `=>` `dx=2du,` `u_l=0/2=0,` `u_l=infty/2=infty` (`u_l` and `u_u` are lower and upper bound respectively). `2int_0^infty sin u...

Math`int_0^infty cos (pi x)dx=` Substitute `u=pi x` `=>` `du=pi dx` `=>` `dx=(du)/pi,` `u_l=pi cdot 0=0,` `u_u=pi cdot infty=infty.` (`u_l` and `u_u` are lower and upper bound respectively)....

Math`int_0^infty e^x/(1+e^x)dx=` Substitute `u=1+e^x` `=>` `du=e^xdx,` `u_l=1+e^0=2,` `u_u=lim_(x to infty)(1+e^x)=infty.` `int_1^infty 1/u du=ln u_2^infty=lim_(u to infty)ln uln 2=infty` As...

Math`int_0^infty 1/(e^x+e^x)dx=` Multiply both numerator and the denominator by `e^x.` `int_0^infty e^x/(1+e^(2x))dx=` Substitute `u=e^x` `=>` `du=e^x dx,``u_l=e^0=1,``u_u=lim_(x to infty)e^x=infty...

Math`int_infty^infty 4/(16+x^2)dx=` Divide both numerator and denominator by 16. `int_infty^infty (1/4)/(1+x^2/16)dx=int_infty^infty (1/4 dx)/(1+(x/4)^2)=` Substitute `u=x/4` `=>` `du=1/4 dx.`...

Math`int_1^infty (ln x)/x dx=` Substitute `u=lnx` `=>` `du=1/x dx,` `u_l=ln 1=0,` `u_u=ln infty=infty` (`u_l` and `u_u` denote lower and upper bound respectively). `int_0^infty u...

Math`int_4^infty 1/(x(ln x)^3)dx=` Substitute `u=ln x` `=>` `du=1/x dx,` `u_l=ln 4,` `u_u=ln infty=infty` (`u_l` and `u_u` denote lower and upper bound respectively). `int_(ln 4)^infty1/u^3...

MathWe will use integration by parts (twice): `int u dv=uvint v du` `int_0^infty e^x cos x dx=[u=e^x,dv=cos x dx],[du=e^x dx,v=sin x]=` `e^x sin x+int_0^infty e^x sin x dx=[u=e^x,dv=sin x...

MathWe will use integration by parts `int u dv=uvint v du` We will need to apply integration by parts two times in order to eliminate `x^2` from under the integral. `int_0^infty x^2e^x...

MathWe will use integration by parts `int udv=uvint vdu` `int_0^infty xe^(x/3)dx=[u=x,dv=e^(x/3)dx],[du=dx,v=3e^(x/3)]=` `3xe^(x/3)_0^infty+3int_0^infty e^(x/3)dx=`...

MathWe will use integration by parts `int udv=uvint vdu` `int_infty^0 xe^(4x)dx=[u=x,dv=e^(4x)dx],[du=dx,v=1/4e^(4x)]=` `1/4xe^(4x)_infty^0+1/4int_infty^0 e^(4x)dx=`...

Math`int_1^infty 4/root(4)(x)dx=4int_1^infty x^(1/4)dx=4x^(3/4)/(3/4)_1^infty=16/3x^(3/4)_1^infty=` `16/3(lim_(x to infty) x^(3/4)1)=16/3(infty1)=infty` As we can see the value if the integral is...

Math`int_1^infty 3/root(3)(x)dx=` Rewrite the surd as a power using the following formula `root(n)(x^m)=x^(m/n).` `int_1^infty 3x^(1/3)dx=3x^(2/3)/(2/3)_1^infty=^(9x^(2/3))/2_1^infty=9/2( lim_(x to...

MathAn integral in which one of the limits of integration is infinity is an improper integral. Because we cannot find the definite integral using infinity (since it is not an actual value), we will...

MathThe function under the integral is continuous and bounded on any interval `[1, A].` The only point that makes the integral improper is `x=+oo.` Therefore this integral is the limit `lim_(A>+oo)...

MathAny integral with infinite bounds is an improper integral therefore this is an improper integral. `int_infty^0 e^(3x) dx=` Substitute `u=3x` `=>` `du=3dx,` `u_l=3cdot(infty)=infty,`...

MathIntegral is improper if we have to take limit in order to calculate it. This can happen if we have infinite values of integration or if the interval if integration contains point(s) where the...

MathThe integral is improper because the function under the integral `f(x)=1/(x3)^(3/2)` is not defined at 3 (for `x=3` denominator is equal to zero). `int_3^4 1/(x3)^(3/2)dx=` Substitute `u=x3`...

MathThe integral is improper because the function under the integral is not defined at zero (see the image below). `1/sqrt0=1/0` `int_0^4 1/sqrt x dx=2sqrt x_0^4=2(sqrt 4sqrt0)=4` The integral...

MathIntegral `int_0^(pi/4)csc x dx` is improper because cosecant is not defined on zero (more generally `csc x` is not defined for `x in {k pi, k in ZZ}.`) and interval of integration includes that...

MathAny integral with infinite bounds is an improper integral, hence integral `int_infty^infty sin x/(4+x^2)dx` is an improper integral. It can be shown that `int_infty^infty sin x/(4+x^2)dx=0` The...

MathAny integral with infinite bounds is an improper integral. Hence `int_0^infty cos x dx` is an improper integral. Also, the cosine function is a periodic function and `lim_(x to infty)cos x` does...

MathThe integral does not have infinite bounds and the function is well defined over the whole interval of integration so there is no need to use limits. Therefore, the integral is not improper....

MathAny integral with infinite bounds is an improper integral, therefore `int_1^infty ln(x^2)dx` is an improper integral. Moreover, since the function under the integral is positive over the given...

MathIntegral `int_1^2 dx/x^3` is not an improper integral because function `f(x)=1/x^3` is well defined (with finite values) over the whole interval of integration `[1,2].` ` ` The only point where the...

MathIntegral is improper if we have to take limit in order to calculate it. This can happen if we have infinite values of integration or if the interval if integration contains point(s) where the...

MathGivne to solve, `lim_(x>1^(+)) (int_1^x cos(theta) d theta ) / (x1)` =`lim_(x>1^(+)) ([sin(theta)]_1^x) / (x1)` =`lim_(x>1^(+)) ([sin(x)sin(1)]) / (x1)` when `x> 1+` then...

Math`lim_(x>oo) (int_1^x ln(e^(4t1)) dt )/ x` = `lim_(x>oo) (int_1^x (4t1) dt )/ x` = `lim_(x>oo) ( [4(t^2)/2t]_1^x )/ x` = `lim_(x>oo) ( [2x^2 x][2(1^2)1] )/ x` = `...

MathGiven to solve, `lim_(x>0) x/arctan(2x)` as `x>0` then the `x/arctan(2x) =0/0` form so upon applying the L 'Hopital rule we get the solution as follows, as for the general equation it is as...